More About This Title Lower Previsions

English

This book has two main purposes. On the one hand, it provides aconcise and systematic development of the theory of lower previsions,based on the concept of acceptability, in spirit of the work ofWilliams and Walley. On the other hand, it also extends this theory todeal with unbounded quantities, which abound in practicalapplications.

Next, we extend the framework of sets of acceptable gambles to consideralso unbounded quantities. As before, we again derive rationalitycriteria and inference methods for lower previsions, this time alsoallowing for conditioning. We apply this theory to constructextensions of lower previsions from bounded random quantities to alarger set of random quantities, based on ideas borrowed from thetheory of Dunford integration.

A first step is to extend a lower prevision to random quantities thatare bounded on the complement of a null set (essentially boundedrandom quantities). This extension is achieved by a natural extensionprocedure that can be motivated by a rationality axiom stating thatadding null random quantities does not affect acceptability.

In a further step, we approximate unbounded random quantities by asequences of bounded ones, and, in essence, we identify those forwhich the induced lower prevision limit does not depend on the detailsof the approximation. We call those random quantities 'previsible'. Westudy previsibility by cut sequences, and arrive at a simplesufficient condition. For the 2-monotone case, we establish a Choquetintegral representation for the extension. For the general case, weprove that the extension can always be written as an envelope ofDunford integrals. We end with some examples of the theory.

English

English

Preface xv

Acknowledgements xvii

1 Preliminary notions and definitions 1

1.1 Sets of numbers 1

1.2 Gambles 2

1.3 Subsets and their indicators 5

1.4 Collections of events 5

1.5 Directed sets and Moore–Smith limits 7

1.6 Uniform convergence of bounded gambles 9

1.7 Set functions, charges and measures 10

1.8 Measurability and simple gambles 12

1.9 Real functionals 17

1.10 A useful lemma 19

PART I LOWER PREVISIONS ON BOUNDED GAMBLES 21

2 Introduction 23

3 Sets of acceptable bounded gambles 25

3.1 Random variables 26

3.2 Belief and behaviour 27

3.3 Bounded gambles 28

3.4 Sets of acceptable bounded gambles 29

3.4.1 Rationality criteria 29

3.4.2 Inference 32

4 Lower previsions37

4.1 Lower and upper previsions 38

4.1.1 From sets of acceptable bounded gambles to lower previsions 38

4.1.2 Lower and upper previsions directly 40

4.2 Consistency for lower previsions 41

4.2.1 Definition and justification 41

4.2.2 A more direct justification for the avoiding sure loss condition 44

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